Global wellposedness of the equivariant Chern–Simons–Schrödinger equation

Abstract

In this article we consider the initial value problem for the $m$-equivariant Chern–Simons–Schrödinger model in two spatial dimensions with coupling parameter $g\in \mathbb{R}$. This is a covariant NLS type problem that is $L^2$-critical. We prove that at the critical regularity, for any equivariance index $m\in \mathbb{Z}$, the initial value problem in the defocusing case ($g<1$) is globally wellposed and the solution scatters. The problem is focusing when $g\ge 1$, and in this case we prove that for equivariance indices $m\in \mathbb{Z}$, $m\ge 0$, there exist constants $c=c_{m,g}$ such that, at the critical regularity, the initial value problem is globally wellposed and the solution scatters when the initial data ${\varphi }_0\in L^2$ is $m$-equivariant and satisfies $\Vert {\varphi }_0{\Vert }_{L^2}^2<c_{m,g}$. We also show that $\sqrt{c_{m,g}}$ is equal to the minimum $L^2$ norm of a nontrivial $m$-equivariant standing wave solution. In the self-dual $g=1$ case, we have the exact numerical values $c_{m,1}=8\pi (m+1)$.